On Quantum Theory of Transport Phenomena

948

Progress of Theoretical Physics, Vol. 20, No.6, December 1958

On Quantum Theory of Transport Phenomena

--Steady Dijfusion--

Sadao NAKAJIMA

Physical Institute, Nagoya University, Nagoya

(Received August 17, 1958)

A general formulation is given to the quantum theory of steady diffusion. In seeking for a steady solution of Liouville's equation, the boundary condition is taken into account by requiring that the solution should lead to a given distribution of average density. The distribution is to be determined by macroscopic law of diffusion and macroscopic boundary condition.

The basic equation thus obtained has a form similar to Bloch's kinetic equation and reduces to the latter in the limit of a system of weakly interacting particles. This is shown by generalizing a damping theoretical expansion of Kohn and Luttinger.

It is found that the Einstein relation is valid only for the symmetric part of diffusion- and electric conductivity tensors, in agreement with Kasuya's suggestion.

§ 1. Introduction

Recently Kubo and others1),2) have succeeded in formulating quantum statistical ex

pressions for transport coefficients such as electric and thermal conductivities. These

formulae are just as general and rigorous as, say, the familiar expression for the partition

function Z=Tr(exp { H/kT}). In practice appropriate approximation should, of course,

be made in evaluating transport coefficients. The point is, however, that the conventional

Bloch equation is nothing else but the lowest order approximation in a damping theoretical

treatment of dynamical motion (see § 6) and by 110 means the most general way of ap

proaching the problem.

Now, in the case of mechanical disturbances such as an external electric field, deri

vation of these formulae has been rather simple. The mechanical disturbance is expressed

as a definite perturbing Hamiltonian and the deviation from equilibrium caused by it can

be obtained by perturbation theory. On the other hand, thermal disturbances such as

density and temperature gradients cannot be expressed as a perturbing Hamiltonian in an

unambiguous way. Accordingly in the previous paper,2) use had to be made of Onsager's

assumption that the average regression of spontaneous fluctuation follows the macroscopic

laws. As a result, certain ambiguity has been left over, concerning. galvano-magnetic

effects caused by thermal disturbances.

In the present paper, a general formulation free from such an assumption will be

given to the theory of . thermal disturbances and, in order to show the basic idea, the case

of steady diffusion will be discussed in detail. In this case, the usual Bloch equation

takes the form in which the drift term due to the density gradient is balanced by the

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On Quantum Theory of Transport Phenomena 949

<collision term. Our purpose is then to generalize the equation so as to include system;;

of strongly interacting particles (electrons, phonons, im puri ties, etc.). In other words, we

shall seek for a steady solution of the quantum theoretical Liouville equation

i[H, /'J 0, (1·1)

where H is the Hamiltonian of our system and /' the density matrix, taking fj = 1. The

boundary condition should be such that a steady gradient of average density is established

by attaching suitable sources at the boundary. But for such a boundary condition, (1· 1)

would lead to an equilibrium density matrix, microcanonical, canonical or grand canonical.

The microscopic detail of the interaction with sources at the boundary, however, should

not be essential to the law of diffusion as an intrinsic property of a large system. We

need only to suppose that the solution of (1· 1) should lead to a given distribution of

average density. The distribution is in turn to be determined by the macroscopic law

of diffusion together with a given macroscopic boundary condition. The law of diffusion

itself is a consequence from our solution of (1. 1), so that the method is self-consistent.

In this sense, the thermal disturbance is a constraint upon dynamical motion and in

fact appears as an effective potential in our basic equation (~4). The equation takes a

form similar to the Bloch equation and reduces to the latter in the limit of a system of

weakly interacting particles. This will be shown by applying a damping theoretical ex

pansion due to Kohn and Luttinger~) (§ 6).

An important consequence of the theory is that the well-known Einstein relation is

valid only for the symmetric part of diffusion- and electric conductivity tensors. In other

words, as regards the Hall effect, the gradient of chemical potential is not equivalent to

the electric field. The difference becomes appreciable at low temperatures and under

strong magnetic fields. Such a difference has first been suggested by Kasuya and is con

firmed far beyond doubt by the present theory (§ 5) . From technical points of view, the present theory is similar to the so-called method

of pseudo-potentia1.4) It may also be regarded as generalization of Enskog's classical kinetic

theory.';) Indeed, Matsubara(») has once tried the theory of hydrodynamic properties from

the latter point of view. It seems, however, that neither physical nor mathematical details

has ever been examined by these previous authors.

§ 2. Local equilibrium

For definiteness, let us take a system of similar particles (say, electrons in a metal),

in which there exists a steady gradient of average density. The number density of

particles is represented by the operator

n(x) = 2.~r)(X-Xj) =J2-1 Y'n"eii.",K', (2·1) j k

where J2 is the volume of the system and n", = 2~ exp { - il~· Xj} should not be confused

with occupation numbers. The average density is then given by

Tr(pn(x» =(n(x». (2·2)

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950 S. Nakajima

As mentioned in § 1, we suppose that the right-hand side of (2·2) is given as a solution of the macroscopic diffusion equation, hence (2· 2) is the condition for the solution of (1· 1) to satisfy.

Let us describe the system in equilibrium by the grand canonical distribution

Pe=exp{~N-{iH} /Tr(exp{~N-(3H}), (2.3)

where ~ is the chemical potential, f9 the temperature and N the operator representing the total number of particles and commutable with th~ Hamiltonian H. We shall assume that the system in equilibrium is homogeneous so that

(2.4)

is ,constant. Hence (2· 3), though stationary, does not satisfy (2· 2) when the density

gradient does exist. A possibility of satisfying (2·2) is given by the so-called local equilibrium distribu-·

tion

Pl=exp {¢+fN- {iH+ 2.j/~kn_k}' (2·5) k::j::O

Here ¢ 1S a normalization constant to make

Tr(pz) =1 (2·6)

and the ~ k (k 0) represent the fluctuation of chemical potential in space. From (2· 6) it can be easily seen that

(2·7)

where the entropy S is defined as

-Tr(p,logp,), (2·8)

Obviously, (2·2) is always satisfied by (2·5) with suitably chosen f k • But (2·5}

does not satisfy (1.1), [H, Pt] ~ O. We are thus led to assume the density matrix in the form

ACtually, however, no approximation is introduced in so far as Writing the density matrix: in this form. For any given p, we can always find (2·5) with parameters ¢, f, (3 and

fk so chosen that

Tr(p) =Tr(p,), Tr(pN) =Tr(Pl N) (2·10)

Then (2· 9) can be regarded as the definition of ,01' which should necessarily satisfy

Tr(P1) =0, Tr(p1N) =0,

Tr(p1H) =0, Tr(P1nk) =0. (2.11)

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There are an infinite number of density matrices which satisfy (2· 10) . Among

them, (2· 5) is characterized by the maximum entropy and makes it possible to introduce.

thermodynamical variables as shown by (2·7). This is the reason for our particular choice (2·5), because the boundary condition of our problem is most simply expressed in

terms of thermodynamical variables by specifying chemical potentials of sources at the

boundary.

§ 3. Linear approximation

Now let us assume that the system is not far from equilibrium so that both ~k and

PI are sman quantities of first order. Then the well-known expansion formula of ordered

exponential leads to

(3 ·1)

where we have made use of (2·4), i. e., •

(3·2)

and n,~(-i;') is obtained from Heisenberg's operator nk(t) =exp(iHt) ·nk·exp( -iHt) by replacing t with - ii. .

Under the linear approximation (3· 1), the average density is given as

where

~

gk= J di. (n_k( ii.)nk\' o

In particular, assuming gk IS continuous at k=o, we can show that

limgk=tJ(an/af) k~O

where n IS defined by (2· 4) . From (2· 7), the entropy is expressed as

S=So- ~ ~'gi/ (nk) (n_,,)

(3·3)

(3 ·4}

(3·5}

(3·6)

with the equilibrium value So. If the density fluctuation is gradual, i. e., only the ~k

with small k are excited, we may replace gk, in (3· 6) by (3·5), so that we obtain the. well-known expansion of entropy in thermodynamics.

Let us now turn to the flow density which is defined by

vex) (3·7)

In cases of neutral particles and also of charged particles without magnetic fields, the.

local equilibrium (3· 1) makes no contribution to the average flow. In the case of charged

particles moving in a magnetic field, on the contrary, the flow does not vanish even in

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'952 S. Nakajima

,equilibrium, because there exists a diamagnetic current. This current, however, can be

described in terms of a magnetization and, in particular, never gives rise to a net current

flowing through a cross section of a conductor (except for superconductors). So we here

after disregard this part of flow.

There still remains the flow due to the second term of (3· 1)

~ d) .\ (3' (n_l',( i)v"')e' o

(3·8)

This vanishes if there is no magnetic field. Because then n", is invariant and Vir. changes

its sign under the transformation of time reversal. In the presence of a magnetic field,

however, the transformation of time reversal includes the reversal of the magnetic field,

too. So we can infer only that (3· 8) is reversed together with the magnetic field. In

other words, the flow is antisymmetric with respect to the magnetic field. But it does

vanish in the classical limit and therefore in the high temperature limit, too. In fact,

in the classical limit, we can first perform the integration over particle velocities in taking

the expectation value in (3· 8) and this vapishes because it is linear in velocities.

A more detailed discussion of (3·8) will be given in § 5.

§ 4. Basic equation and its solution

Now inserting (2·9) together with (3· 1) into (1· 1), we obtain our basic equation

f\

i[H, ('1J=- )}{"J ~: Pe~-l.(-i) o

where, of course,

A particular solution of the inhomogeneous equation (4 -1) IS given by

CD (.

PI = .\ dt e-5t e- iHt [' eiHt

()

(4 ·1)

(4·2)

(4-3)

where [' stands for the right-hand side of (4. 1) . In fact, (4· 3) is the solution which

we are seeking for. This can be seen in the following way.

First, note that the macroscopic relaxation time, i. e., the relaxation time in which

the system recovers equilibrium, is of the order of [2/ D. Here I is the linear dimension

of the system and D the diffusion coefficient. Now suppose that the system was in

equilibrium at the remote past, t = - 00, and that the difference of chemical potentials

at two ends of the system has been increased very slowly from zero to the present value

(t=O). For instance, suppose that it is proportional to exp(Ct), where o <c<t,D/l2.

Then we may assume that a steady diffusion is established at each instant of time from

,f = - 00 to t = O. This is a sort of adiabatic change and may be represented by the

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~olution proportional to expect) of the following Liouville equation

At t=O, we have

(4·4)

whose solution is nothing but (4· 3), as can be easily confirmed.

It still remains to be confirmed that (4· 3) does satisfy our subsidiary conditions

(2 ·11). For any dynamical va.riable a, we have

co ~1 ..

-V"" If' et \" d;. . ~ :;1.0, dt e- . 9 < n_k ( () 0 I

Tr(Pl a) = . (4·5)

If we take a=l, the left-hand side is Tr(pJ, a.nd the right-hand side contains <~-k\~' This should vanish, because any flow in equilibrium is steady. The first condition in

(2 . 11) is thus satisfied. As for the second condition, we take a = N. Since N com

mutes with the Hamiltonian, the right-hand side of (4·5) contains < ~-k N) c" This

should also vanish, because

Tr(e~N-:~H~ .. N) -I.

Thus the second condition is satisfied. Similarly, it can be seen by taking a = H that

.the third condition is also satisfied.

As for the final condition in (2· 11), we have

00 ;~

. f' f'dJ. Tr(P1n,.) = -il1,f",. J dt e-St

.\ T (V_h( -iA)n",(t»e o 0

(4·6)

where we have made use of the continuity equation (4·2). In general, (4· 6) does not

vanish. We should, therefore, restrict ourselves to the case where the density fluctuation

is so gradual that we may replace ( .. -) e in (4· 6) by < V N) e • Here V is the net flow,

1. e., v'" with k 0. Since there is no net flow in equilibrium as mentioned before, we

have

Thus, under our restriction, the fourth condition in (2 ·11) is satisfied.

The restriction can more precisely be expressed in the following way. In terms of

space coordinates, (4· 6) is written as

Tr(P1n(x» - J dx' R(x-x') ·J7;(x' ) (4·7)

with

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954 S. Nakajima

00 ~ d}' R(x-x') = 1 dt e-f-t 1 Q (v (x', -i}.)n(x, t) >e.

o 0 t-'

Except for such singular systems as superconductors and superfluids, the relaxation function

R (x) will decay off within a finite distance. The gradient of chemical potential' Ii' ~ should be nearly constant over this distance. This is the precise formulation of our

restriction. If the condition is not satisfied, the simple themodynamical notion of diffu

sion is not applicable in describing th~ phenomenon.

§ 5. Diffusion constant and Einstein's relation

Now, we take a=Vk in (4·5), then the flow arising from PI is given as

co ~

(Vk) 1 ==Tr(P1Vk) = -~k i dt e-Et i ~ (~-k( -i}.)Vk(t) >e. () 0

(5 ·1)

Making use of the continuity equation (4· 2), we see that in the limit of k -7> 0 the

diffusion equation takes the form

('Ok(J.)l = - 2JDJ~) X (ik.., ~k)' (5·2) "\I

fl., lJ = x, y, z.

In terms of space coordinates, this is written as

(5·3)

Here the diffusion tensor Dl;{J- is given by

co ~,

DJ~) = i dt e-,-t 1 ~ (V"\I( -iA)V(J.(t) >e () ()

(5 ·4)

where V is the net flow.

On the .other hand, it has been foundl) that the electric conductivity tensor IS given,.

in general, by

CfJ ~

0- ILl; = 1 dt e-Et 1 dA (Iv (- iA)ItJ. (t) >e o 0

where J = e V is the electric current and e the charge of the particle.

with (5· 5), we obtain the well-known Einstein relation

o-wlJ=pe2 D~~).

(5 '5)

Comparing (5· 4 )

(5·6)

It should be remembered, however, that in the presence of a magnetic field we have the

flow arising from the local equilibrium distribution, (3· 8), which is antisymmetric and

thus makes contribution to the antisymmetric part of the diffusion tensor. Thus we.

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obtain the important conclusion that the Einstein relation is valid only for the symmetric

part of diffusion and electric conductivity tensors.

In order to see the situation in more detail, let us take a system of electrons in a

metal. For simplicity we shall neglect the interaction with phonons, taking account of

impurity scattering alone. The Hamiltonian then takes the form

H= 22H' (X j ) , j

where A is the vector potential of the magnetic field and cp the scalar potential including

both periodic and impurity potentials. Hereafter an operator of one electron will be

primed as H'. Let us introduce the system of one electron eigenfunctions

Then the Hamiltonian can be written as

where ar , ar * are destruction and creation operators of electrons In the r-state.

where

Remember that

and also that

nk = 2J (rlnk'i s) ar * ao

Vk= 2i(rlvk'ls)a,.*aH

~=i[H', x].

(as * a,.ar! * as! ) e = as.S! arr! f( E8 ) (1- f( E,.) )

where f is the Fermi distribution function

Then it is easily seen that (5·1) can be written as

P

(5·7)

Similarly

(5·8)

(5·9)

(5 ·10)

(5·11)

where P indicates taking the principal value. The flow arising from this part is anti

symmetric, whereas the flow arising from the a.function is symmetric.

Similarly, (3·8) can be written as

(5·12)

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956 S. Nakajima

For the sum of those terms in which E,. ~ Ep we can make use of

(sl~~klr) ices E .. ) (sln~,.,lr) (5 ·13)

so that the sum exactly cancels the anti symmetric part of (5· 11) , On the other hand,

the terms with E.. in (5, 12) can be transformed as

From (5·9) we see

x ~ {(sln~klr) (rlv,/Is) (slvk'lr) (rln~kls)}.

, , ) - " V'" n_k -x

(5·14 )

so that the first term of (5· 14) vanishes. For the second term, we can again make

use of (5· 13) . Taking the limit of k-~ 0 we find the anti symmetric part of the diffusion tensor

D (A)fL).I-

On the other hand, the antisymmetric palt of the electric conductivity tensor is given as

1 (T(A) !,I..\I (slxILlr) (rlx).Ils)}.

(5·16)

Now, characteristic frequencies of the electron velocity (slxlr)exp{i(E, E,.)t} are

cyclotron frequency We, collision frequency ,and also the average interval of interband

transitions LlE. If all these frequencies satisfy

(5' 17)

then the difference quotient of f in (5 ·16) IS practically the same as the differential

quotient in (5· 15), and we have the Einstein relation for the anti symmetric part, too.

It is to be noted here that the second of (5· 17) is the well-known criterion for ap

plicability of Bloch's kinetic equation, although actually this is too stringent.

§ 6. Damping theoretical expansion

Finally, we shall show that our basic equation (4· 1) reduces to the usual Bloch

equation in the limit of a system of weakly interacting particles. In fact (4·1) has

mathematically the same form as discussed by Kohn and Luttingerg) in the case of a

steady electric field. They have dealt with a simple model in which an electron is

scattered by impurities. Actually their method can be applied to integration of the

Liouville equation in general, provided that th~ Hamiltonian satisfies certain conditions.

First let us introduce a symbolic method due to Kubo. Define the linear operator

J;, which operates always from the left on any dynamical variable 7) as

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~.~==i[H, ~J. (6·1)

Incidentall y, the operator of this sort has been introduced in classical statistical mechanics

to prove the ergodic theorem.7) The Liouville equation can be written as

a at

Let us introduce the Laplace transform

co f'

{i(s) .\dte-stp(t) o

which converges for ffis>o. Then (6·2) is transformed into

(s+ ~)p(s) =Po

(6-2)

(6·3)

(6·4)

where ,()o is the initial density matrix at t = o. Replacing P (s) by PI' Po by r, and

taking the limit S -7> + 0, we obtain our basic equation (4·4). But we shall be concerned

with the general case (6· 4) for the moment.

Now, assume that the Hamiltonian takes the form

H=Ho+gHr (6·5)

where g is a small numerical parameter indicating the order of perturbation H J • Cor-

respondingly the operator splits as

(6·6)

Let us fix the basic state vectors I a) as

In the case of the electron-phonon interaction, for instance, a stands for a set of occupa

tion numbers of free electrons and phonons. We introduce linear operators 5) and X which also operate from the left on any dynamical variable as

(al 5)~la') = (al~la) (~('t('t!'

(a I X ~ 1 at) = (a 1 11 a') (1- l; ",,! ) .

Without loss of generality we assume that 5) HI =0. Following Kohn and Luttinger, we

decompose the density matrix into diagonal and non-diagonal parts

pes) (5) +('n(s), (6 -7)

Inserting (6 -7) together with (6·6) into (6·4) and taking diagonal and non-diag.:mal parts of the equation respectively, we obtain

s/'r1(s) +g5)~JPn(s) =5)/,0' (6·8)

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958 S_ Nakajima

Eliminating {in from these, we have a formal solution for the diagonal part

We can derive a power series expansion from (6 -9), supposing that

and making use of

In particular the lowest order approximation satisfies

Remember that 0:>

----1-1)= r dtexp( st) -exp( iHot) -1) exp(iHot)_ s+ J

o

Then, taking the explicit matrix representation of (6 - 12), . we find

25 -----1 (a 1 HI I a') \2 (P~' (s) - p~ (s) ) 52+ «(V~_(V~,)2

In the limit of s ~ + 0, we have

(6-10)

(6-11)

(6-12)

(6-13)

Hence (6 -13) is nothing else but the Laplace transform of the so-called master equation

P~(t) = ~27rg21(a\Hlla')120'«(V~_(V~') (P(J.,(t) -P~(t»,

P~(t=O) =(alpola)-

which van Hove8) has obtained by me-.at;ls of rather a lengthy expansion_

Now replacing /' by Pu Po by I', and taking the limit s ~ + 0, we have

21'( g2~ I (aIH1Ia') \26 (W(J.-(V(J.f) (FcfJ - F~) = (alI'la') ~,

(6 -14)

(6·15)

where F~ is the diagonal element of ,01 in the lowest order approximation. The equation

(6 -15) says that the drift term due to the gradient of chemical potential is b~lanced by

the collision term due to perturbation HI' More precisely, F~ is still a many~particle

distribution function and we have further to reduce (6· 15) to obtain the Bloch equation

of one particle distribution function_

Of course, certain conditions should be satisfied in order that the transport equation

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,( 6 ·12) be already a good approximation. Smallness of the parameter g is by no means

: sufficient. For instance, in the case of an electron scattered by impurities, the contribu

tion of the second term in (6· 11) relative to the first would not be simply proportional

to g, but to g fJ, if impurities were on regular lattice points. In this case the transport

equation can never be a good approximation, because gJJ can be increased indefinitely by

increasing the volume fJ. In general, the size dependence of perturbation in a large

· system plays an . essential role here. We shall not enter into this prl)blem, as the detailed

·analyses have been given by van Hove.S)

§ 7. Conclusions

Quantum statistical mechanics of steady diffusion has been formulated. The theory

-is the most natural generalization of the Bloch kinetic equation and in fact reduces to

the latter under certain conditions. An important conclusion is that the Einstein relation

is valid only for the antisymmetric part of diffusion- and electric conductivity tensors.

Obviously, the theory can be generalized so as to include heat conduction and viscosity,

,which will be discussed in a subsequent paper.

The expressions for transport coefficients derived in this way are general and rigorous.

· In statistical thermodynamics, we have the general expression for the partition function

-and introduce approximate methods in evaluating this; the virial expansion in the case

of imperfect gases, normal vibrations in the case of crystals, and so on. In just the same

. way, we should introduce appropriate methods of approximation to evaluate transport

· coefficients, starting with our general expressions. Thus we may couclude that quantum

-.theory of transport coefficients now stands on the same level as statistical mechanics of

equilibrium properties, though admittedly we know few methods of approximation such as

Bloch's kinetic equation which is nothing but the lowest order approximation in a damp

ing theoretical expansion.

References

1) H. Nakano, Prog. Theor. Phys. 15 (1956), 77. R. Kubo, J. Phys. Soc. Japan 12 (1957), 570.

2) R. Kubo, M. Yokota, and S. Nakajima, J. Phys. Soc. Japan 12 (1957), 1203. 3) W. Kohn and J. M. Luttinger, Phys. Rev. 108 (1957), 590.

·4) H. B. Callen, M. L. Barasch and J. L. Jackson, Phys. Rev. 88 (1952), 1382.

R. Kubo, N. Hashitsume and M. Yokota, Busseiron Kenkyu, 91 (1955), 32 .

. 5) S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases., Cambridge, 1939.

6) T. Matsubara, Busseiron Kenkyu, vol. 2 (1957), 190.

_7) E. Hopf, Ergodentheorie, Berlin, 1937.

-8) L. van Hove, Physica 21 (1955), 517; ibid. 23 (1957), 441.

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On Quantum Theory of Transport Phenomena